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Limits without log

  • Thread starter wimma
  • Start date
  • #1
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Homework Statement


Find the limit as n--> infinity of (1-2/n)^n


Homework Equations



We know (1+1/x)^x --> e as n--> infinity

The Attempt at a Solution



I worked it out as e^(-2) using log but I can't get it out using the fundamental limit above. I know it's the square of (1-1/x)^x (where we let x=n/2), just I don't know how to show that (1-1/x)^x --> 1/e. If you could let x |--> -x somehow I'd get the desired result using the limit laws but I'm not sure that's allowed.
 

Answers and Replies

  • #2
33,181
4,860
Let n = -2x. This makes your limit
[tex]\lim_{-2x \to \infty} (1 + \frac{1}{x})^{-2x}[/tex]

With a bit of adjustment you can use the limit you know.
 
  • #3
38
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but won't the parameter go to -infinity so we can't equate (1+1/x)^x to e?
 
  • #4
33,181
4,860
As it turns out,
[tex]\lim_{x \to -\infty} (1 + \frac{1}{x})^x~=~e[/tex]

Can you use this fact?
 

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